# Automorphic L-function

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Mathematical concept

In [mathematics](/source/Mathematics), an **automorphic *L*-function** is a function *L*(*s*,π,*r*) of a complex variable *s*, associated to an [automorphic representation](/source/Automorphic_representation) π of a [reductive group](/source/Reductive_group) *G* over a [global field](/source/Global_field) and a finite-dimensional complex representation *r* of the [Langlands dual group](/source/Langlands_dual_group) *L**G* of *G*, generalizing the [Dirichlet L-series](/source/Dirichlet_L-series) of a [Dirichlet character](/source/Dirichlet_character) and the [Mellin transform](/source/Mellin_transform) of a [modular form](/source/Modular_form). They were introduced by [Langlands](/source/Robert_Langlands) ([1967](#CITEREFLanglands1967), [1970](#CITEREFLanglands1970), [1971](#CITEREFLanglands1971)).

[Borel (1979)](#CITEREFBorel1979) and [Arthur & Gelbart (1991)](#CITEREFArthurGelbart1991) gave surveys of automorphic L-functions.

## Properties

Automorphic L {\displaystyle L} -functions should have the following properties (which have been proved in some cases but are still conjectural in other cases).

The L-function L ( s , π , r ) {\displaystyle L(s,\pi ,r)} should be a product over the places v {\displaystyle v} of F {\displaystyle F} of local L {\displaystyle L} functions.

L ( s , π , r ) = ∏ v L ( s , π v , r v ) {\displaystyle L(s,\pi ,r)=\prod _{v}L(s,\pi _{v},r_{v})}

Here the [automorphic representation](/source/Automorphic_representation) π = ⊗ π v {\displaystyle \pi =\otimes \pi _{v}} is a tensor product of the representations π v {\displaystyle \pi _{v}} of local groups.

The L-function is expected to have an analytic continuation as a meromorphic function of all complex s {\displaystyle s} , and satisfy a functional equation

L ( s , π , r ) = ϵ ( s , π , r ) L ( 1 − s , π , r ∨ ) {\displaystyle L(s,\pi ,r)=\epsilon (s,\pi ,r)L(1-s,\pi ,r^{\lor })}

where the factor ϵ ( s , π , r ) {\displaystyle \epsilon (s,\pi ,r)} is a product of "local constants"

ϵ ( s , π , r ) = ∏ v ϵ ( s , π v , r v , ψ v ) {\displaystyle \epsilon (s,\pi ,r)=\prod _{v}\epsilon (s,\pi _{v},r_{v},\psi _{v})}

almost all of which are 1.

## General linear groups

[Godement & Jacquet (1972)](#CITEREFGodementJacquet1972) constructed the automorphic L-functions for general linear groups with *r* the standard representation (so-called [standard L-functions](/source/Standard_L-function)) and verified analytic continuation and the functional equation, by using a generalization of the method in [Tate's thesis](/source/Tate's_thesis). Ubiquitous in the Langlands Program are [Rankin-Selberg](/source/Rankin%E2%80%93Selberg_method) products of representations of GL(m) and GL(n). The resulting Rankin-Selberg L-functions satisfy a number of analytic properties, their functional equation being first proved via the [Langlands–Shahidi method](/source/Langlands%E2%80%93Shahidi_method).

In general, the [Langlands functoriality](/source/Langlands_functoriality) conjectures imply that automorphic L-functions of a connected [reductive group](/source/Reductive_group) are equal to products of automorphic L-functions of general linear groups. A proof of Langlands functoriality would also lead towards a thorough understanding of the analytic properties of automorphic L-functions.

## See also

- [Grand Riemann hypothesis](/source/Grand_Riemann_hypothesis)

## References

- Arthur, James; [Gelbart, Stephen](/source/Stephen_Gelbart) (1991), "Lectures on automorphic L-functions", in Coates, John; Taylor, M. J. (eds.), [*L-functions and arithmetic (Durham, 1989)*](http://www.claymath.org/library/cw/arthur/pdf/automorphic-L.pdf) (PDF), London Math. Soc. Lecture Note Ser., vol. 153, [Cambridge University Press](/source/Cambridge_University_Press), pp. 1–59, [doi](/source/Doi_(identifier)):[10.1017/CBO9780511526053.003](https://doi.org/10.1017%2FCBO9780511526053.003), [ISBN](/source/ISBN_(identifier)) [978-0-521-38619-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-38619-7), [MR](/source/MR_(identifier)) [1110389](https://mathscinet.ams.org/mathscinet-getitem?mr=1110389)

- [Borel, Armand](/source/Armand_Borel) (1979), "Automorphic L-functions", in [Borel, Armand](/source/Armand_Borel); Casselman, W. (eds.), [*Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2*](https://www.ams.org/publications/online-books/pspum332-index), vol. XXXIII, Providence, R.I.: [American Mathematical Society](/source/American_Mathematical_Society), pp. 27–61, [doi](/source/Doi_(identifier)):[10.1090/pspum/033.2/546608](https://doi.org/10.1090%2Fpspum%2F033.2%2F546608), [ISBN](/source/ISBN_(identifier)) [978-0-8218-1437-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8218-1437-6), [MR](/source/MR_(identifier)) [0546608](https://mathscinet.ams.org/mathscinet-getitem?mr=0546608)

- Cogdell, James W.; Kim, Henry H.; Murty, Maruti Ram (2004), [*Lectures on automorphic L-functions*](https://books.google.com/books?id=jb3ZCp0-MQsC), Fields Institute Monographs, vol. 20, Providence, R.I.: [American Mathematical Society](/source/American_Mathematical_Society), [ISBN](/source/ISBN_(identifier)) [978-0-8218-3516-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8218-3516-6), [MR](/source/MR_(identifier)) [2071722](https://mathscinet.ams.org/mathscinet-getitem?mr=2071722)

- [Gelbart, Stephen](/source/Stephen_Gelbart); Piatetski-Shapiro, Ilya; Rallis, Stephen (1987), *Explicit Constructions of Automorphic L-Functions*, Lecture Notes in Mathematics, vol. 1254, Berlin, New York: [Springer-Verlag](/source/Springer-Verlag), [doi](/source/Doi_(identifier)):[10.1007/BFb0078125](https://doi.org/10.1007%2FBFb0078125), [ISBN](/source/ISBN_(identifier)) [978-3-540-17848-4](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-17848-4), [MR](/source/MR_(identifier)) [0892097](https://mathscinet.ams.org/mathscinet-getitem?mr=0892097)

- [Godement, Roger](/source/Roger_Godement); Jacquet, Hervé (1972), *Zeta Functions of Simple Algebras*, Lecture Notes in Mathematics, vol. 260, Berlin, New York: [Springer-Verlag](/source/Springer-Verlag), [doi](/source/Doi_(identifier)):[10.1007/BFb0070263](https://doi.org/10.1007%2FBFb0070263), [ISBN](/source/ISBN_(identifier)) [978-3-540-05797-0](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-05797-0), [MR](/source/MR_(identifier)) [0342495](https://mathscinet.ams.org/mathscinet-getitem?mr=0342495)

- Jacquet, H.; Piatetski-Shapiro, I. I.; Shalika, J. A. (1983), "Rankin-Selberg Convolutions", *Amer. J. Math.*, **105** (2): 367–464, [doi](/source/Doi_(identifier)):[10.2307/2374264](https://doi.org/10.2307%2F2374264), [JSTOR](/source/JSTOR_(identifier)) [2374264](https://www.jstor.org/stable/2374264)

- Langlands, Robert (1967), [*Letter to Prof. Weil*](http://publications.ias.edu/rpl/section/21)

- Langlands, R. P. (1970), "Problems in the theory of automorphic forms", [*Lectures in modern analysis and applications, III*](http://publications.ias.edu/rpl/section/21), Lecture Notes in Math, vol. 170, Berlin, New York: [Springer-Verlag](/source/Springer-Verlag), pp. 18–61, [doi](/source/Doi_(identifier)):[10.1007/BFb0079065](https://doi.org/10.1007%2FBFb0079065), [ISBN](/source/ISBN_(identifier)) [978-3-540-05284-5](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-05284-5), [MR](/source/MR_(identifier)) [0302614](https://mathscinet.ams.org/mathscinet-getitem?mr=0302614)

- Langlands, Robert P. (1971) [1967], [*Euler products*](http://publications.ias.edu/rpl/paper/37), Yale University Press, [ISBN](/source/ISBN_(identifier)) [978-0-300-01395-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-300-01395-5), [MR](/source/MR_(identifier)) [0419366](https://mathscinet.ams.org/mathscinet-getitem?mr=0419366)

- Shahidi, F. (1981), "On certain "L"-functions", *Amer. J. Math.*, **103** (2): 297–355, [doi](/source/Doi_(identifier)):[10.2307/2374219](https://doi.org/10.2307%2F2374219), [JSTOR](/source/JSTOR_(identifier)) [2374219](https://www.jstor.org/stable/2374219)

v t e L-functions in number theory Analytic examples Riemann zeta function Dirichlet L-functions L-functions of Hecke characters Automorphic L-functions Selberg class Algebraic examples Dedekind zeta functions Artin L-functions Hasse–Weil L-functions Motivic L-functions Theorems Analytic class number formula Riemann–von Mangoldt formula Weil conjectures Analytic conjectures Riemann hypothesis Generalized Riemann hypothesis Lindelöf hypothesis Ramanujan–Petersson conjecture Artin conjecture Algebraic conjectures Birch and Swinnerton-Dyer conjecture Deligne's conjecture Beilinson conjectures Bloch–Kato conjecture Langlands conjecture p-adic L-functions Main conjecture of Iwasawa theory Selmer group Euler system

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